- #1
Ed Quanta
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If S is a closed surface, then the integral over S of (curlV) dot dn must equal zero.
How could I show this is true in general?
How could I show this is true in general?
Stoke's Theorem is a mathematical theorem that relates the surface integral of a vector field over a surface to the line integral of the same vector field along the boundary of the surface.
Stoke's Theorem is significant because it allows for the calculation of surface integrals, which are often difficult to solve, through the use of line integrals, which are typically easier to solve.
Stoke's Theorem is applicable when the surface is bounded by a simple, closed, and piecewise-smooth curve and the vector field is continuously differentiable on the surface.
Stoke's Theorem is a higher-dimensional version of the Fundamental Theorem of Calculus. While the Fundamental Theorem of Calculus relates the integral of a function over an interval to its derivative, Stoke's Theorem relates the surface integral of a vector field to its line integral.
Stoke's Theorem is used in many areas of science and engineering, such as fluid dynamics, electromagnetics, and geophysics. It is used to calculate quantities such as fluid flow and electric and magnetic fields over surfaces, which have practical applications in fields such as aerospace and energy production.